Hard Drive Swap, Cant Save to it unless I run software as admin

Hi!
So like I have this annoying problem which I hope someone can help me with. So basically Mm old computer died, I bought a new one, took the hard drive out of the old one and put it in the new one as a second hard drive. Now my problem is, whenever I want to save to that hard drive from any software, whether I'm rendering video to it, or simply just saving a document to it, I keep getting a message saying something like, I don't have the privileges or rights to save to that location. However if i run the software, for example notepad, as an admin and then try and save to that hard drive, it works and if i save to the first hard drive and then copy to the second one, I'm able to do that too. So! is there a simple fix for this rather than having to reformat the whole thing? I'm guessing its something to do with the windows ownership/user accounts as it came from a different computer.